Finite Group Databasebeta

← #31913721

Subgroups — #31913721

Order 18176.

642 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588268C212yesmaximal
2#22586467C214yesmaximal, complemented
3C71×#2297C211yesmaximal, characteristic
4C2×#14556047C2214yes—
5C2×#14556045—22no—
6#17985963—22no—
7C22×#41×C71C2211yescharacteristic
8C2×#117×C71C411yescharacteristic
9C22×C42×C71C411yescharacteristic
10#391×C71C2212yes—
11#390×C71C412yes—
12#386×C71—22no—
13C4×C22⋊C4×C71—22no—
14C2×#11765849—24no—
15#14556047—28no—
16#14556045—44no—
17C22⋊(C71⋊C8)—44no—
18C2×#41×C71—24no—
19C2×#41×C71—42no—
20C2×#41×C71—42no—
21C22×Q8×C71C2311yescharacteristic
22C22×D4×C71C2×C411yescharacteristic
23C22×D4×C71D412yes—
24C23×C4×C71—21no—
25C23×C4×C71C2×C411yescharacteristic
26C23×C4×C71D412yes—
27#117×C71—42no—
28C2×C4⋊C4×C71C2×C412yes—
29C2×C22⋊C4×C71—21no—
30C2×C22⋊C4×C71—24no—
31C2×C42×C71—21no—
32C2×C42×C71—21no—
33C2×C42×C71—42no—
34C2×C42×C71C2×C412yes—
35#95×C71—24no—
36#11765849—44no—
37#11765849—44no—
38C2×C71⋊C8—44no—
39C24×C71C22⋊C412yes—
40C24×C71C22⋊C411yescharacteristic
41#41×C71—416no—
42C2×Q8×C71—41no—
43C2×Q8×C71—41no—
44C2×Q8×C71#4114yes—
45C2×D4×C71—24no—
46C2×D4×C71—22no—
47C2×D4×C71—24no—
48C2×D4×C71—41no—
49C2×D4×C71—41no—
50C2×D4×C71—42no—
51C22×C4×C71—22no—
52C22×C4×C71—22no—
53C22×C4×C71—22no—
54C22×C4×C71—22no—
55C22×C4×C71—24no—
56C22×C4×C71—24no—
57C22×C4×C71—22no—
58C22×C4×C71—42no—
59C22×C4×C71—44no—
60C22×C4×C71—42no—
61C22×C4×C71C22⋊C411yescharacteristic
62C22×C4×C71C22×C411yescharacteristic
63C22×C4×C71C2×D412yes—
64C4⋊C4×C71—42no—
65C22⋊C4×C71—48no—
66C22⋊C4×C71—42no—
67C42×C71—42no—
68C42×C71—42no—
69C42×C71—42no—
70C23×C71—21no—
71C23×C71—21no—
72C23×C71—22no—
73C23×C71—22no—
74C23×C71—22no—
75C23×C71—42no—
76C23×C71—44no—
77C23×C71#9912yes—
78C23×C71C2×C22⋊C411yescharacteristic
79Q8×C71—44no—
80D4×C71—48no—
81D4×C71—44no—
82C2×C4×C71—28no—
83C2×C4×C71—21no—
84C2×C4×C71—21no—
85C2×C4×C71—21no—
86C2×C4×C71—21no—
87C2×C4×C71—22no—
88C2×C4×C71—22no—
89C2×C4×C71—24no—
90C2×C4×C71—48no—
91C2×C4×C71—42no—
92C2×C4×C71—48no—
93C2×C4×C71—42no—
94C2×C4×C71—44no—
95C2×C4×C71—42no—
96C2×C4×C71#9912yes—
97C2×C4×C71#11712yes—
98C2×C4×C71#12314yes—
99C71⋊C8—84no—
100C22×C71—22no—
101C22×C71—22no—
102C22×C71—42no—
103C22×C71—42no—
104C22×C71—44no—
105C22×C71—42no—
106C22×C71—42no—
107C22×C71—44no—
108C22×C71—82no—
109C22×C71#38111yescharacteristic
110C22×C71C2×#9911yescharacteristic
111C22×C71C2×#9911yescharacteristic
112C4×C71—24no—
113C4×C71—22no—
114C4×C71—22no—
115C4×C71—44no—
116C4×C71—82no—
117#12861—711nomaximal, Sylow
118C2×C71—21no—
119C2×C71—21no—
120C2×C71—42no—
121C2×C71—82no—
122C2×C71#181911yescharacteristic
123C2×C71#213212yes—
124#2297D7111yescharacteristic
125#1785—712no—
126#1394—714no—
127C71#1286111yescharacteristic, complemented, Sylow
128C22×#41C2×D7111yescharacteristic
129C2×#117Dic7111yescharacteristic
130C22×C42Dic7111yescharacteristic
131C2×#101—714no—
132C2×#100—1422no—
133#407—1422no—
134#391C2×D7112yes—
135#390Dic7112yes—
136#386—22no—
137C4×C22⋊C4—22no—
138C2×#41—24no—
139C2×#41—42no—
140C2×#41—42no—
141C22×Q8C22×D7111yescharacteristic
142C22×D4C2×Dic7111yescharacteristic
143C22×D4C22⋊D7112yes—
144C23×C4—21no—
145C23×C4C2×Dic7111yescharacteristic
146C23×C4C22⋊D7112yes—
147C2×M16—1424no—
148#117—42no—
149C2×C4⋊C4C2×Dic7112yes—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—24no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—42no—
155C2×C42C2×Dic7112yes—
156#101—1428no—
157#100—2844no—
158C22⋊C8—2844no—
159#95—24no—
160C24C22⋊Dic7112yes—
161C24C22⋊Dic7111yescharacteristic
162#41—416no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8#1176586514yes—
166C2×D4—24no—
167C2×D4—22no—
168C2×D4—24no—
169C2×D4—41no—
170C2×D4—41no—
171C2×D4—42no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—42no—
180C22×C4—44no—
181C22×C4—42no—
182C22×C4C22⋊Dic7111yescharacteristic
183C22×C4C22×Dic7111yescharacteristic
184C22×C4C2×C22⋊D7112yes—
185M16—2844no—
186M16—2844no—
187C2×C8—2844no—
188C4⋊C4—42no—
189C22⋊C4—48no—
190C22⋊C4—42no—
191C42—42no—
192C42—42no—
193C42—42no—
194C23—21no—
195C23—21no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—42no—
200C23—44no—
201C23#1455604312yes—
202C23C2×C22⋊Dic7111yescharacteristic
203Q8—44no—
204D4—48no—
205D4—44no—
206C2×C4—28no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—48no—
215C2×C4—42no—
216C2×C4—48no—
217C2×C4—42no—
218C2×C4—44no—
219C2×C4—42no—
220C2×C4#1455604312yes—
221C2×C4#1455608712yes—
222C2×C4#1455611114yes—
223C8—5684no—
224C22—22no—
225C22—22no—
226C22—42no—
227C22—42no—
228C22—44no—
229C22—42no—
230C22—44no—
231C22—42no—
232C22—82no—
233C22#1798586211yescharacteristic
234C22C2×#1455604311yescharacteristic
235C22C2×#1455604311yescharacteristic
236C4—24no—
237C4—22no—
238C4—22no—
239C4—44no—
240C4—82no—
241C2—21no—
242C2—21no—
243C2—42no—
244C2—82no—
245C2#2258841411yescharacteristic
246C2#2259002512yes—
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC22×Q8C22×D71
FittingC71×#2297C2
socleC22×C71C2×#99
prime radicalC22×#41×C71C22
cosocleC22×Q8×C71C23
hypercenter#2297D71
hypocenterC71#12861
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2297

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22586467C2semidirect
127. C71#12861semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225882682
#225864672
C71×#22972
#1286171